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MATH0013 Complex Analysis Autumn 2019 Alexander V. Sobolev D epartment of M athematics , U niversity C ollege L ondon , G ower S treet , L ondon , WC1E 6BT E-mail address : [email protected]
2000 Mathematics Subject Classification. . A bstract . This is is a draft a one term course Complex Analysis for the 2nd year.
Contents Chapter 1. Geometry, topology and analysis in the complex plane 1 1. A Review of Complex Numbers 1 2. Geometry, topology in the Complex Plane 5 3. Functions, their limits and continuity 7 Chapter 2. Derivatives and holomorphic functions 11 1. Derivatives 11 2. The Cauchy-Riemann Equations 13 3. Properties of di ff erentiable functions 17 4. Harmonic Functions 19 Chapter 3. Power series and examples of holomorphic functions 21 1. Complex series 21 2. Power series 22 3. The exponential and trigonometric functions 25 4. The Logarithm Function and Branches 28 Chapter 4. Conformal mappings 31 1. Paths 31 2. Conformal mapping 31 3. Examples of conformal maps 33 Chapter 5. Contour Integration and Cauchy’s Theorem 37 1. Paths and Contours 37 2. Path Integrals, antiderivatives 38 3. The Cauchy-Goursat Theorem 43 4. The Cauchy Integral Formula and its consequences 48 Chapter 6. Roots and singularities 59 1. Generalised Power Series: Laurent expansion 59 2. Proof of Theorem 6.1 64 3. Zeros 66 4. Counting roots and poles 69 5. Evaluation of Real Integrals 71 i
CHAPTER 1 Geometry, topology and analysis in the complex plane 1. A Review of Complex Numbers 1.1. Basic facts. Let z = ( x , y ) be a point on the plane R 2 , or, in other words, vectors on the plane. From Linear Algebra we know how to add vectors and multiply them by real-valued constants: z 1 + z 2 = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) , az = ( ax , ay ) , a R . We are going to call these vectors complex numbers , and name the coordinates x and y the real and imaginary part of z : x = Re z , y = Im z . The question: why do we want these new names for the familiar objects? Answer: because we introduce a new operation – multiplication. D efinition 1.1. Define the product of two complex numbers z 1 = ( x 1 , y 1 ), z 2 = ( x 2 , y 2 ) as follows: z 1 z 2 = ( x 1 x 2 - y 1 y 2 , x 1 y 2 + y 1 x 2 ) . Note that z 1 z 2 = z 2 z 1 and z 1 ( z 2 + z 3 ) = z 1 z 2 + z 1 z 3 ! Notation for the set of all complex numbers: C . When we view them as points on the plane, we call it the Argand plane . Consider the product of two equal numbers z 1 = z 2 = (0 , 1): (0 , 1) 2 = ( - 1 , 0) . So, if we denote i = (0 , 1) and x = ( x , 0), then we have the familiar identity i 2 = - 1. Now, using the same notation we can use another, more standard way of writing the complex numbers: z = ( x , y ) = x (1 , 0) + y (0 , 1) = x + iy . This is called the canonical, or standard form of complex numbers. We are going to use this form from now on. It is more convenient for remembering how to multiply complex numbers. The inverse number z - 1 = 1 z is defined to be the complex number such that zz - 1 = 1. It is easy to check that z - 1 = x - iy x 2 + y 2 = x x 2 + y 2 - i y x 2 + y 2 .

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